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WhenEvent is working. Try WhenEvent[x[t] < 0, Print[t]; x[t] -> 0] to see that every crossing is detected. The problem is that it only detects crossings. So changing x[t] -> 0 does not reset the event. At the next step x[t] becomes negative and no event is detected. (This is how it is supposed to behave.)
The way to deal with this is to use ...

Mathematica seems to split the integrand component,
E^(-((-m + Log[x])^2/(2 s^2)))
into
E^(-((m^2 + Log[x]^2)/(2 s^2)))
times the sort-of "coefficient"
E^((m Log[x])/s^2) (* == x^(m/s^2) *)
in order to calculate the integral in terms of Meijer $G$. For reasons that are obscure to me, it seems to want the coefficient of m in the exponent to be ...

Since your integrand does not approach zero but a finite positive number,
Limit[Exp[-16.136 (1 - Exp[-0.012*t])], t -> Infinity]
(* 9.82255*10^-8 *)
the integral over {t, 0, Infinity} does not converge.
By the way, the error in the NIntegrate[integrand, {t, 0, 1000}] should be about 10^-7, which seems better than R. In fact, the precision seems ...

(Reposting my comment as an answer, just to take the question off the unanswered list)
This is a bug and has been fixed in version 10.2. I am not aware of any workarounds that may be applicable in earlier versions.

A small remark on the error:
CoefficientArrays::poly: -(1 + 3.27432/(1 + 0.092 Ccu)^2) Ccu11498 - (3 Ccu11499)/200 + 1.38465 Ccu$11500 is not a polynomial. >>
First note that Ccu is your dependent variable in your PDE. The expression being complained about has several variables, Ccu and ones like Ccu$11498.. (The ones like Ccu$11498 are internal, ...

When you try to NIntegrate your expression, the error messages include: "suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small". Your expression does not have a singularity, its integral is manifestly not zero, it is not oscillatory, so it must be a numerical precision problem.
...

It's a good idea to include the error you got in your question. It gives a clue and might prompt someone to investigate:
NDSolve::ndinid: Initial condition {0} is not in the range specified by the discrete variable NDSolve`s$147246. >>
Now Sign is discontinuous and will cause NDSolve to invoke special processing of the ODE. I suspect that the strange ...

Mathematica's finite element PDE methods only work for linear PDEs as of v10.1. See here or here for some other tips on how to deal with this. Given the form of your equation (some kind of non-linear reaction-diffusion equation?), I would suspect that you're better off using Mathematica's default "Method of Lines" algorithm than trying to use finite ...

As a modified version of Michael E2 answer:
I tried to rewrite your original curve as below for $t\in [0,\tfrac{\pi}{2}]$, just in order to make sure the derivatives of the parametric form can be obtained easily by avoiding Abs or Sign:
ncurve={(Cos[t]^2 )^(1/4),(Sin[t]^2)^(1/4)}
Then the result of the closed area can be obtained by applying Greens ...

After thinking about it, i solved the problem by considering:
region = RegionBoundary[
BoundaryDiscretizeRegion[Ellipsoid[{0, 0, 0}, {1, 0.125, 0.125}],
MaxCellMeasure -> 0.1]]
in this only the surface is discretized and the mesh quality can be specified by MaxCellMeasure.

The problem appears to be associated with interpolation by the InterpolatingFunction produced by NDSolve rather than by NDSolve itself. For instance, with
sol = NDSolve[{(R'[t])^2 + 2 R[t] R''[t] == -1, R[1] == 1,
R'[1] == 2/3}, {R}, {t, 1, 1.2}, WorkingPrecision -> 100];
a Plot of R and its first two derivatives yields near t = 0
Plot[{R[t], ...

The OP's -- oops, they're bbodfrey's -- pictures suggest the problem is with interpolation, as bbgodfrey also observed. Some of the problem can be ameliorated with the InterpolationOrder option.
From InterpolationOrder:
In functions such as NDSolve, InterpolationOrder->All specifies that the interpolation order should be chosen to be the same as the ...

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